Maximum principles for a family of nonlocal boundary value problems

BOUNDARY VALUE PROBLEMS

Received 21 October 2003 and in revised form 16 February 2004

We study a family of three-point nonlocal boundary value problems (BVPs) for ann th-order linear forward difference equation. In particular, we obtain a maximum principle and determine sign properties of a corresponding Green function. Of interest, we show that the methods used for two-point disconjugacy or right-disfocality results apply to this family of three-point BVPs.

1. Introduction

The disconjugacy theory for forward difference equations was developed by Hartman [15] in a landmark paper which has generated so much activity in the study of

differ-ence equations. Sturm theory for a second-order finite differdiffer-ence equation goes back to Fort [12], which also serves as an excellent reference for the calculus of finite differences. Hartman considers thenth-order linear finite difference equation

Pu(m)=

n

j=0

αj(m)u(m+j)=0, (1.1)

αnα0=0,m∈I= {a,a+ 1,a+ 2,...}. To illustrate the analogy of (1.1) to annth-order ordinary differential equation, introduce the finite difference operator∆by

∆u(m)=u(m+ 1)−u(m), ∆0_u(m)≡u(m),

∆i+1_u(m)=∆∆i_{u(m), i≥1.} (1.2)

Clearly,Pcan be algebraically expressed as annth-order finite difference operator. Letm1,bdenote two positive integers such thatn−2≤m1< b. In this paper, we as-sume thata=0 for simplicity, and we consider a family of three-point boundary condi-tions of the form

u(0)=0,...,u(n−2)=0, um1

=u(b). (1.3)

Clearly, the boundary conditions (1.3) are equivalent to the boundary conditions

∆i_{u(0)=0, i=0,...,n−2, um}

1

=u(b). (1.4)

There is a current flurry to study nonlocal boundary conditions of the type described by (1.3). In certain sectors of the literature, such boundary conditions are referred to as multipoint boundary conditions. Study was initiated by Il’in and Moiseev [16,17]. These initial works were motivated by earlier work on nonlocal linear elliptic boundary value problems (BVPs) (see, e.g., [3,4]). Gupta and coauthors have worked extensively on such problems; see, for example, [13,14]. Lomtatidze [18] has produced early significant work. We point out that Bobisud [5] has recently developed a nontrivial application of such problems to heat transfer. For the rest of the paper, we will use the term nonlocal boundary conditions, initiated by Il’in and Moiseev [16,17].

We motivate this paper by first considering the equation

Pu(m)=∆n_{u(m)=0, m=0,...,b−n. (1.5)}

In this preliminary discussion, we employ the natural family of polynomials, m(k)₌ m(m−1)···(m−k+ 1) so that∆m(k)_=km(k−1)_.

which are equivalent to the two-point conjugate conditions [15]

u(0)=0,...,u(n−2)=0, u(b)=0. (1.9)

Atm1=b−1, we have the boundary conditions

which are equivalent to the two-point “in between conditions” [9]

u(0)=0,...,u(n−2)=0, ∆u(b−1)=0. (1.11)

The following inequalities have been previously obtained [10,15]:

0> G(n−2;m,s)> G(b−1;m,s), (1.12)

(m,s)∈ {n−1,...,b} × {0,...,b−n}.

The following theorem is obtained directly from the representation (1.6) ofG(m1; m,s).

Theorem1.1. G(m1;m,s)is decreasing as a function ofm1; that is,

0≥Gm1;m,s> Gm1+ 1;m,s

, (1.13)

(m1,m,s)∈ {n−2,...,b−2} × {n−1,...,b} × {0,...,b−n}. The first inequality is strict, except in the conjugate case,m1=n−2, atm=b.

The purpose of this paper is to obtainTheorem 1.1for a more general finite difference

equation,Pu(m)=0. Note that even for the specific BVP (1.5), (1.3), the calculations to show thatGis decreasing inm1 are tedious. The method exhibited in the next section allows one to bypass the tedious calculations. We will need to assume a condition that implies disconjugacy. We will then argue that similar results are obtained if the nonlocal boundary condition has the form

∆j_um

1

=∆j_{u(b−j), j∈ {0,...,n−1}. (1.14)}

The similar results will be valid if we assume thatPu(m)=0 is right-disfocal [2].

2. A general disconjugate equation

Hartman [15] defined the disconjugacy of (1.1) on I = {0,...,b}. First recall the definition of a generalized zero [15].m=0 is a generalized zero ofuifu(0)=0.m >0 is a generalized zero ofuifu(m)=0, or there exists an integerk≥1 such thatm−k≥0, u(m−k+ 1)= ··· =u(m−1)=0, and (−1)ku(m−k)u(m)>0. Then (1.1) is discon-jugate onI ifuis a solution of (1.1) onI and thatuhas at leastngeneralized zeros on I implies thatu≡0 onI. A condition related to and stronger than disconjugacy is that of right-disfocality [1,8]; (1.1) is right-disfocal onI ifuis a solution of (1.1) onI and that∆juhas a generalized zero atsj, 0≤s0≤s1≤ ··· ≤sn−1≤b−n+ 1, implies that u≡0 onI. For this particular paper, a concept of right (n−1;j) disfocality would be appropriate; (1.1) is right (n−1;j) disfocal onI ifuis a solution of (1.1) onIand that uhas at least n−1 generalized zeros ats0,...,sn−2,∆juhas a generalized zero at sn−1,

max{s0,...,sn−2} ≤sn−1≤b−j, imply thatu≡0 onI.

exist positive functionsvidefined on{0,...,b−i+ 1}such that

We will remind the reader of a version of Rolle’s theorem.

Lemma2.1. Letube a sequence of reals defined on a set of integers. IfPjuhas generalized zeros atµ1andµ2, whereµ1< µ2, thenPj+1uhas a generalized zero in{µ1,...,µ2−1}. Proof. Hartman [15] proved thatvj+2Pj+1uhas a generalized zero in the set{µ1,...,µ2−

1}. The lemma follows sincevj+2is positive.

Theorem2.2. Assume thatPu=0is right(n−1; 1)disfocal on{0,...,b}. Then there exists a uniquely determined Green functionG(m1;m,s)for the BVPs (1.1), (2.3).

Proof. Letvdenote the solution of the initial value problem (IVP) (1.1), satisfying initial conditions

Pjv(0)=0, j=0,...,n−2, Pn−1v(0)=1. (2.4)

Let χ(m,s) denote the Cauchy function for (1.1); that is,χ, as a function of m, is the solution of the IVP (1.1), with the initial conditions

ForceGto satisfy the nonlocal conditionP0(m1)u(m1)=P0(b)u(b); in particular, solve algebraically fora(m1;s) to obtain

am1;s= −P0χ(b,s) P0v(b)−P0v

m1

, m1≤s,

am1;s=P0χ

m1,s−P0χ(b,s) P0v(b)−P0v

m1

, s+ 1≤m1.

(2.7)

Note that the right (n−1; 1) disfocality implies thatP0v(b)−P0v(m1) is nonzero; in particular,a(m1;s) is well defined. Straightforward calculations show that

b−n

s=0

Gm1;m,sf(s) (2.8)

is the unique solution of a nonhomogeneous BVP of the form Pu(m)= f(m), m∈ {0,...,b−n}, satisfying the boundary conditions (2.3).

Theorem2.3. Assume thatPu=0is right(n−1; 1)disfocal on{0,...,b}. Then

Gm1;m,s≤0, (2.9)

(m1,m,s)∈ {n−2,...,b−1} × {n−1,...,b} × {0,...,b−n}. The inequality is strict, ex-cept in the conjugate case,m1=n−2, atm=b.

Remark 2.4. We consider a specific set of nonlocal boundary conditions in this paper to illustrate that theory and methods from disconjugacy theory apply to families of nonlocal BVPs. Because of the specific nonlocal boundary conditions, it is the case thatP1uhas a generalized zero in{m1,...,b−1}. Hence, the argument we produce below is precisely the general argument for the conjugate boundary conditions given in [6, Section 8.8], after Rolle’s theorem has been applied one time.

Proof. It is known that (2.9) is valid in the extreme cases,m1=n−2 [15] andm1=b−1 [10]. Letm1∈ {n−1,...,b−2}be fixed. We first show thatGis of fixed sign for

(m,s)∈ {n−1,...,b} × {0,...,b−n}. (2.10)

Lets∈ {0,...,b−n}be fixed. By construction,G, as a function ofm, satisfies the bound-ary conditions (2.3).

Assume for the sake of contradiction thatGhas an additional generalized zero atm0 for somem0∈ {n−1,...,b}. Then P0G takes on an additional generalized zero atm0 sincev1is of strict sign. Perform a count on the number of generalized zeros of eachPjG.

Note, from the boundary conditions, that P1G has a generalized zero atn−3. The first point of the argument is to argue thatP1Ghas two more generalized zeros in{n− 2,...,b−1}. First assume thatm0≤m1. Apply Rolle’s theorem,Lemma 2.1. ThenP1G has two more generalized zeros,m11< m12, wherem11∈ {n−2,...,m0−1}andm12∈ {m1,...,b−1}. Second, assume thatm1< m0. With this assumption,P0G(m1)=0. Apply Rolle’s theorem to see thatP1Ghas a generalized zero atm11∈ {n−2,...,m0−1}. Ifm11< m1, then Rolle’s theorem can be applied to the nonlocal conditions to obtain a second generalized zerom12∈ {m1,...,b−1}. So, we come to the last subcase,m1≤m11. Assume without loss of generality thatm0is the smallest generalized zero ofP0Gto the right of m1. ThenP0G(m1)P0G(m0)≤0. This implies thatP0G(m0)P1G(m0−1)≥0. These two inequalities imply thatP1Ghas a generalized zero in{m0,...,b−1}. If not, then∆P0G has a fixed sign on{m0,...,b−1}, which agrees with the sign ofP1Gatm0−1. Recall the identity

P0G(b)=P0Gm0+

b−1

µ=m0

∆P0G(µ). (2.11)

In particular,P0G(m1) andP0G(b) have opposite signs which contradicts the nonlocal boundary conditions. Thus, there exists m∈ {m0,...,b−1} such that P1G(m0− 1)P1G(m)≤0. In particular, there existsm12∈ {m0,...,b−1}such thatP1Ghas a gener-alized zero atm12.

To summarize the purpose of the preceding paragraphs, we have shown thatP1Ghas at least three generalized zeros on{n−3,...,b−1}. It now easily follows by induction and repeated applications ofLemma 2.1and the boundary conditions that for each j= 0,...,n−2PjGhas at least 3 generalized zeros, one atn−(j+ 2) and other two satisfying n−(j+ 1)≤mj1< mj2≤b−j.

SincePn−2G has at least three generalized zeros,Pn−2G has at least two generalized zeros counting multiplicities form≤sor fors+ 1≤m. Either case will provide a contra-diction.

Assume thatPn−2Ghas at least two generalized zeros counting multiplicities form≤s. ThenPn−1Ghas at least one generalized zero form≤s. By construction,PnG≡0 fort≤s; thus,vnPn−1Gis of constant sign and has a generalized zero; in particular,Pn−1G≡0 for m≤s. Continue inductively and argue thatPjG≡0 form≤s. In particular,G=v≡0

form≤s. This clearly contradicts the construction ofv.

Assume thatPn−2Ghas at least two generalized zeros counting multiplicities fors+ 1≤ m. Then a similar argument gives thatG=v+χ≡0 fors+ 1≤m. Ifv= −χ, then the disconjugacy is violated.

Thus,Gis of strict sign on{n−2,...,b−1} × {n−1,...,b} × {0,...,b−n}. To determine the sign ofG, evaluate the sign of

h(m)=

b−n

s=0

which is the unique solution of the BVP

Pu=1, m∈ {0,...,b−n}, (2.13)

with boundary conditions (2.3).Pjhhas a generalized zero atn−(j+ 2) because of the

boundary conditions. In addition, because of the nonlocal boundary conditions and re-peated applications of Rolle’s theorem,Pjhhas a generalized zero atmj,1, where

n−(j+ 2)< mj,1< mj−1,1< b (2.14)

forj=1,...,n−2. Moreover, due to Rolle’s theorem,Pn−1hhas precisely one generalized zero sincePnu≡1.

Pnu≡1 implies thatvnPn−1uis increasing. From the above construction,vnPn−1uhas precisely one generalized zero at 0< mn−1,1. Hence, vnPn−1u <0 on{0,...,mn−1,1−1}. Continue inductively. Initially,vn−1Pn−2uis decreasing andPn−2u(0)=0; soPn−2u(1)<0. Inductively, it follows thatPju(n−1−j)<0, j=0,...,n−2. In particular,u(n−1)<0;

sinceGdoes not change sign,udoes not change sign. Thus,unegative implies that (2.9)

is valid.

Theorem2.5. Assume thatPu=0is right (n−1; 1)disfocal on{0,...,b}. ThenG, as a function ofm1, is decreasing; that is, ifn−2≤m1< m2≤b−1, then

Gm2;m,s< Gm1;m,s≤0, (2.15)

(m,s)∈ {n−1,...,b} × {0,...,b−n}. The second inequality is strict, except in the conjugate case,m1=n−2, atm=b.

To prove the above comparison theorem, we first obtain a useful lemma. LetG2denote the quasidifference ofGwith respect tom; that is, let

G2

m1;m,s=P0G

m1;m+ 1,s−P0G

m1;m,s=v2P1G. (2.16)

Lemma2.6. Letm1∈ {n−2,...,b−2}.Then G2

m1+ 1;m1,s

<0, s∈ {0,...,b−n}. (2.17)

Proof. The proof requires only a simple extension from the proof ofTheorem 2.3. As summarized in the fourth paragraph of the proof ofTheorem 2.3, we know thatP1Ghas precisely one generalized zerom11to the right ofn−3. We also know by Rolle’s theorem thatm1≤m11.

G(n−2)=0, G(n−1)<0 imply that P1G(n−2)<0 which in turn implies that

P1G(m1−1)<0.

Proof ofTheorem 2.5. LetG1denote the difference ofGwith respect tom1; that is, let G1

m1;m,s=Gm1+ 1;m,s

−Gm1;m,s. (2.18)

Note thatG1is the unique solution of the BVP

Pu=0, m∈ {0,...,b−n}, Pju(0)=0, j=0,...,n−2, P0u(b)−P0u

m1

=G2

m1+ 1;m1,s

<0.

(2.19)

G1 satisfies the difference equation and each of the initial conditions at 0; this is clear since each term ofG(m1+ 1;m,s),G(m1;m,s) satisfies the difference equation and the initial conditions. Simply calculate the nonlocal boundary condition

P0Gm1+ 1;b,s−Gm1;b,s−P0Gm1+ 1;m1,s−Gm1;m1,s =P0Gm1+ 1;b,s−P0Gm1+ 1;m1+ 1,s

+P0G

m1+ 1;m1+ 1,s

−P0G

m1+ 1;m1,s

=G2

m1+ 1;m1,s

.

(2.20)

In particular,

P0u(b)< P0um1. (2.21)

The boundary conditions at 0 and the right (n−1; 1) disfocality imply thatP0uis mono-tone form > n−2.P0u(b)< P0u(m1) implies thatP0uis monotone-decreasing and (2.15)

is proved.

We end the paper with a brief general observation. Letl∈ {0,...,n−2}. Letm1∈ {n−2,...,b−l−1}. Consider the BVP

Pu(m)=0, m∈ {0,...,b−n}, (2.22)

with boundary conditions

Pju(0)=0, j=0,...,n−2, Plum1=Plu(b−l). (2.23)

We state without proof theorems analogous to Theorems2.3and2.5. The observation to make now is that the BVP atl=0,m1=b−1 is equivalent to the BVP withl=1, m1=n−3. One can now begin an inductive argument onland repeat the arguments in the paper.

A Green functionG(l,m1;m,s) for the BVP (1.1), (2.23) is readily constructed as in the proof ofTheorem 2.2. So, from the above observation, we claim that

G(0,b−1;m,s)=G(1,n−3;m,s). (2.24)

Define the jth difference ofGwith respect tomby∆jG. The proof ofTheorem 2.3

gen-eralizes readily to show that

(m1,m,s)∈ {n−l−2,...,b−l−1} × {n−l−1,...,b−l} × {0,...,b−n}. We do not present the proofs because the arguments, applied to∆lG(l,m1;m,s), go through in

com-plete analogy; the inequalities for the lower-order differences are then obtained through repeated definite summations fromm=0, which are valid because of the boundary con-ditions.

Theorem2.7. Assume thatPu=0is right(n−1;l)disfocal on{0,...,b}. Then

∆jGl,m1;m,s≤0, (2.26)

(m1,m,s)∈ {n−2,...,b−l−1} × {n−j−1,...,b−j} × {0,...,b−n}, j=0,...,l. The inequality is strict, except in the casej=l,m1=n−l−2, atm=b−l.

Theorem2.8. Assume thatPu=0is right(n−1;l)disfocal on{0,...,b}. Then∆jG, as a function ofm1, is decreasing; that is, ifn−2≤m1< m2≤b−l−1, then

∆jGl,m2;m,s<∆jGl,m1;m,s≤0, (2.27)

(m,s)∈ {n−j−1,...,b−j} × {0,...,b−n}, j=0,...,l. The second inequality is strict, except in the case j=l,m1=n−l−2, atm=b−l.

Finally, in the spirit of the interesting comparison theorems first introduced by Elias [7] (see also [19] or [11]) and later discretized [10], we close with the following compar-ison theorem.

Theorem2.9. Assume thatPu=0is right-disfocal on{0,...,b}. Letl1< l2.Then

∆jGl2,ml2;m,s<∆jGl1,ml1;m,s≤0, (2.28)

(m,s)∈ {n−j−1,...,b−j} × {0,...,b−n},j=0,...,l1. The inequality is strict, except in the casej=l1,m1=n−l1−2, atm=b−l1. Ifl1=l2andm1< m2, then∆jG(l,m2;m,s)<

∆jG(l,m1;m,s)≤0, (m,s)∈ {n− j−1,...,b−j} × {0,...,b−n}, j =0,...,l1 =l2. The inequality is strict, except in the case,j=l1,m1=n−l1−2, atm=b−l1.

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